We give a positive answer to a problem of Lindenstrauss by showing that the family of compact Hausdorff spaces which are homeomorphic to weakly compact subsets of Banach spaces (Eberlein compacts) is stable under continuous images. This is equivalent to the fact that a Banach space E is a subspace of a WCG space iff the unit ball of E * is an Eberlein compact when equipped with the wMopology. We also study some topological properties of Eberlein compacts.

Xi+ 2i F»-+2, Zi+Zy Fi+3, Xi+Z' (G) The plane Yi+ 2 Zi +2 Ui separates Xi and Z,-+i from X i+2 , Xi+z, Yi+z, and Zi+ Z. (H) The plane XiZi+ 2 Ui separates Y i+2 and Z i+ x from Y if Z if and Zi+z.

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Our method using CH is a blend of two earlier constructions (Hajnal-Juhász [2] and Ostaszewski [4]) of hereditarily separable (HS), regular, non-Lindelöf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under ◇, which implies the continuum hypothesis (CH) but is not equivalent to it.